From which topic should I start learning mathematics?

In school teachers taught us mathematics only for the sake of passing the board exams, due to that I always found mathematics boring. Recently I came accross some books which really made me realize the beauty of mathematics. Now I want to self study Mathematics from basics and cover undergraduate level mathematics in the next 4-5 years.
I am an Electrical & Electronic Engineering fresher and I can occasionally get help from my Maths lecturers.

What topics are covered in undergraduate level Mathematics and in what order are they covered?

Which books should I go through to cover any gaps in basics?

Which topics and books should I start with?
Please suggest good books for all the topics.

Is the syllabus out lined in this page "http://hbpms.blogspot.com/" [Broken] proper and correct?

Please add any suggestions and tips.

I need everything to be planned out and ready before I start anything hence I desperately need these answers, otherwise I can't proceed. I have looked all over internet but couldn't find any clear answer.

Use the internet for basics.
The best way to learn math is to start with the bits that draws your attention and backfill as you go. It's a bit like learning a language - you learn best by living in a country where the language is normally spoken all the time. As far as college math is concerned - they will have a course for you. Just do it.

You need:
arithmetic (can you add and multiply and divide by hand?)
numbers: you need to be able to manipulate fractions and negative numbers.
algebra - balancing equations.

from there you can build quite a tool-box.
You can get all these from geometry but its probably easier this way around.

apart from that - take advise from people who liked math at school with a grain of salt. It's probably fine it's just that they think different from you ... recall: they liked math. Nuff said :)

The biggest shock you'll get is that the sort of problems you will get when you go on further with maths won't involve you just mechanically sticking things into algorithms or having to work out sums of big numbers in your head- it's more about logic, concepts and figuring out how to unwrap sometimes quite complex definitions for yourself instead of being sure that you don't make mistakes when manipulating large formulas. People often overlook this and get the impression that mathematicians are just good for being able to do big sums/long formulas. It's much more creative and interesting than that!

In that respect, I recommend most strongly getting to grips with logic, proofs and the flow of valid mathematical arguments. Mathematical papers are a series of definitions, lemmas, theorems and proofs. You need to make sure that you understand the language of mathematics and understand the concept of a valid proof. Make sure that you understand the concepts of the contrapositive, proof by contradiction, negation, proof by induction, quantifiers, (and tonnes more proof/logic concepts...), try to understand some simple proofs e.g. that sqrt(2) is irrational or that there are infinitely many primes, and try having a go at some proofs for yourself.

I'd say that stuff is the most important. When you've got that down- introduce yourself to real analysis, linear algebra and group theory.

You can't get past school level without it! Of course it's important! You need to know how to add and multiply before you go into college level!

The biggest shock you'll get is that the sort of problems you will get when you go on further with maths won't involve you just mechanically sticking things into algorithms or having to work out sums of big numbers in your head- it's more about logic, concepts and figuring out how to unwrap sometimes quite complex definitions for yourself instead of being sure that you don't make mistakes when manipulating large formulas. People often overlook this and get the impression that mathematicians are just good for being able to do big sums/long formulas. It's much more creative and interesting than that!

I'll second this.

In that respect, I recommend most strongly getting to grips with logic, proofs and the flow of valid mathematical arguments. Mathematical papers are a series of definitions, lemmas, theorems and proofs. You need to make sure that you understand the language of mathematics and understand the concept of a valid proof. Make sure that you understand the concepts of the contrapositive, proof by contradiction, negation, proof by induction, quantifiers, (and tonnes more proof/logic concepts...), try to understand some simple proofs e.g. that sqrt(2) is irrational or that there are infinitely many primes, and try having a go at some proofs for yourself.

That would be the "learn the grammar and the vocab then learn the language" approach.
Exactly the opposite of what I was advocating: the "immersion" method.

In school teachers taught us mathematics only for the sake of passing the board exams, due to that I always found mathematics boring. Recently I came accross some books which really made me realize the beauty of mathematics. Now I want to self study Mathematics from basics and cover undergraduate level mathematics in the next 4-5 years.
I am an Electrical & Electronic Engineering fresher and I can occasionally get help from my Maths lecturers.

What topics are covered in undergraduate level Mathematics and in what order are they covered?

Which books should I go through to cover any gaps in basics?

Which topics and books should I start with?
Please suggest good books for all the topics.

Is the syllabus out lined in this page "http://hbpms.blogspot.com/" [Broken] proper and correct?

Please add any suggestions and tips.

I need everything to be planned out and ready before I start anything hence I desperately need these answers, otherwise I can't proceed. I have looked all over internet but couldn't find any clear answer.

Thank you for your time and valuable suggestion.

the blog you linked to has the right idea. but...it's just a guide, everyone's path is different. for example, some people take a shine to analysis, and get all excited discussing measureability and diffeomorphisms, or that weird continuous function that isn't differentiable anywhere. some people like algebra, or number theory, and abhor the sight of an integral sign, or a rational function in z. others like the pure formalism of logic, languages and symbols, and the intricacies of the "subatomic structure of math".

you'll have to discover for yourself, what things you prefer.

calculus is a fairly basic place to start, it's a first-year undergraduate class in most universities. if you find calculus rough going, then back-track and start with "pre-calculus" or even "college algebra" (note: do NOT confuse that with abstract algebra, or what is often just called "algebra", which is a more advanced subject).

if you're already familiar with calculus, you have a lot of freedom in deciding where to go next. you can study...still more calculus (there's about 2-3 years of material, if you really want to know it well). the higher phases of calculus are often called by different names, because of the wide variety of topics available (multivariate calculus, calculus of variations, real analysis, integration and measure theory, just to name a few). related to this is complex analysis, which is similar to multivariate calculus, but has its own unique flavor, due to some singular properties that complex numbers have.

you'll probably want to tackle linear algebra, as well, as (along with calculus) it is one of the "core" subjects, every mathematician knows (at least a little of). again, the well is deep, you can skim it, and just learn the basics of matrices, and the key concepts of vectors and vector spaces, or you can go deeper into dual spaces, tensor analysis, spectral decompositions, all sorts of fun stuff. you might spend as little as a few months, or 2-3 years here (there's lots of interesting things that can be vectors, including some interesting kinds of functions).

another "basic" class is abstract algebra, which covers a lot of ground as well. after doing math for a while, you'll see certain kinds of structures come up again and again. this field (does that count as a pun?) covers structures in their abstract form (a sort of "let's handle all the cases at once" sort of thing). i recommend having some calculus and a little linear algebra before-hand, just so you have some depth of experience, but other people will tell you "dive right in".

and there's more: differential equations, differential geometry, topology, homotopy and homology, category theory, computability theory, statistics, combinatorics, graphs and trees,it's a long list, and it's growing all the time. you won't be able to learn it all, try a little of everything, and gorge yourself on what you enjoy.

while it's good to have a plan, keep in mind, you'll likely change your mind about some things along the way. keep your options open, there's no one "right way" to learn math.

Use the internet for basics.
The best way to learn math is to start with the bits that draws your attention and backfill as you go. It's a bit like learning a language - you learn best by living in a country where the language is normally spoken all the time. As far as college math is concerned - they will have a course for you. Just do it.

You need:
arithmetic (can you add and multiply and divide by hand?)
numbers: you need to be able to manipulate fractions and negative numbers.
algebra - balancing equations.

from there you can build quite a tool-box.
You can get all these from geometry but its probably easier this way around.

apart from that - take advise from people who liked math at school with a grain of salt. It's probably fine it's just that they think different from you ... recall: they liked math. Nuff said :)

I do know my arithmetic and numbers.. :)
I was talking about gaps in algebra...
thanks for your advice..

You can't get past school level without it! Of course it's important! You need to know how to add and multiply before you go into college level

I'm not saying that it isn't important to be able to add and multiply, of course everyone needs to be able to do that. What I'm saying is that being good at it isn't as important (or important at all) at college level- I'm currently (trying) to do mathematical research and my mental arithmetic is really pretty bad.

If the OP was 7 years old, then I'd probably be suggesting that he/she learns to add and multiply, it's an important step, but I was under the impression that this was already a skill that they'd acquired given that they are asking what they should learn to cover college level maths.

And clear guidance as to which subjects should be studied first, and in which order, I think is quite important. You will gain much more studying things in a sensible order (at least, when beginning college level maths). There is a student my friend has supervised who is far ahead of the others, in first year, with a topological and metric spaces book. My friend told him that although it's good to look ahead, there are plenty of subjects that would be better suited for someone just beginning, not only to help them learn the topic faster, but often to put it into context ("enrich, not accelerate" is a nice phrase). Learning real analysis, for example, before metric spaces would seem very sensible.

Telling the OP just to "immerse himself", in my opinion, isn't all that helpful. It's pretty much non-advice- to someone who doesn't know much about mathematics, how are they to pick what they should study first? How are they to know whether or not it will be helpful to them, or is fundamental or specialist?

At a higher level, I'd agree with you. Perhaps a mix of immersion and direct guidance is best. However, when beginning college level maths, I really recommend picking the right subjects to study first. It's not a coincidence that most courses on mathematics all cover the same basic material in the first couple of years.

The biggest shock you'll get is that the sort of problems you will get when you go on further with maths won't involve you just mechanically sticking things into algorithms or having to work out sums of big numbers in your head- it's more about logic, concepts and figuring out how to unwrap sometimes quite complex definitions for yourself instead of being sure that you don't make mistakes when manipulating large formulas. People often overlook this and get the impression that mathematicians are just good for being able to do big sums/long formulas. It's much more creative and interesting than that!

In that respect, I recommend most strongly getting to grips with logic, proofs and the flow of valid mathematical arguments. Mathematical papers are a series of definitions, lemmas, theorems and proofs. You need to make sure that you understand the language of mathematics and understand the concept of a valid proof. Make sure that you understand the concepts of the contrapositive, proof by contradiction, negation, proof by induction, quantifiers, (and tonnes more proof/logic concepts...), try to understand some simple proofs e.g. that sqrt(2) is irrational or that there are infinitely many primes, and try having a go at some proofs for yourself.

I'd say that stuff is the most important. When you've got that down- introduce yourself to real analysis, linear algebra and group theory.

And it's not the only thing you'll want to be doing when starting mathematics. As I said, the most important things to focus on are logic/proofs, real analysis and calculus, linear algebra and perhaps some group theory/abstract algebra. How to study these though, and possibly what books might be helpful to you, hopefully others can tell you.